2. MITOS Y MITOLOGÍAS POLÍTICAS
2.5 Siglo XX: caos en el Caribe
2.5.3 La Edad de Oro: los gloriosos tiempos de España
(4.5.2)
(4.5.3)
This means the inlet loss coefficient, , varies between 0.5 for sharp inlets to 0.15 for well rounded inlet.
There is also another sort of added friction near the pipe inlet. The Darcy-Weisbach friction factor can be around 2.5 times as high in the first diameter length of pipe compared to what it is in stabilized flow. This effect rapidly diminishes as the velocity profile reaches its normal shape some 20-30 diameters downstream (Idelchik, 1992). It is difficult to quantify this extra loss accurately, and it is common to neglect it, but it is still worth keeping in mind that the total inlet loss is somewhat higher than predicted by equation 4.5.2 or 4.5.3.
4.6 Diameter changes
When the diameter changes, the friction losses become higher than they would have been in a straight pipe.
d r
v e
Figure 4.6.1. Conical diffuser.
Just like for the pipe inlet discussed in chapter 4.5, the friction loss is not the only mechanism at work here: The Bernoulli energy equation tells us that when the velocity is reduced, such as in a diffuser, some of the kinetic energy is transformed to pressure energy.
If the flow had been lossless, the pressure change for incompressible flow in a diffuser would be: of a Francis or Kaplan water turbine: The diameter is increased behind the turbine to extract as much of the kinetic energy from the flowing water as possible. But in addition to this energy transformation, there is also an energy loss due to increased friction. That loss has the opposite effect on .
In general, very small angles lead to a smooth flow with relatively small losses, and the flow follows the conical geometry without separating from the wall. Increasing β beyond a certain point leads to separation, and the losses increase. At exactly which angle separation starts depends on both Re,
and any upstream disturbances.
Measurements carried out by Idelchik (1992) indicate that if βc ≤ 20, no separation occurs under any circumstances, and losses are kept to a minimum. For relatively large βc, separation becomes so dominant that the conical section has no effect, and one may as well use an abrupt diameter step (βc = 900). Crane‟s simplified correlations take this into account, and are adequate for most engineering applications:
* (
) + (4.6.2)
v
outv
in βc* (
) +
Figure 4.6.2. Loss coefficient for conical diffuser.
Note that Kf is defined according to the velocity in the smallest diameter section, the inlet, so that:
(4.6.3)
If we compare the pressure gain described by equation 4.6.1 with the empirical friction losses modeled by equation 4.6.2, we will see that which one is largest depends on the diffuser‟s construction – it is obviously possible to construct diffusers where the pressure increases and diffusers where it is reduced.
Figure 4.6.3. Conical contraction.
Figure 4.6.4. Loss coefficient for conical
V
inV
out βc
contraction.
On equation form, the curves in figure 4.6.1 can be written as:
* (
) +
* (
) + √
(4.6.4)
In this case, too, the Bernoulli-effect also contributes, but both the acceleration and the friction lead to pressure reductions.
4.7 Junctions
Friction losses in pipe junctions are somewhat different from the other losses discussed so far in that junctions are characterized by very many different variables. The geometries may differ in various ways, such as angles, cross sections, and even number of branches. The flow situation also plays a role, such as how the flow is distributed between the different inlet(s) and outlet(s), and whether the junction is used to merge or split flows. Needless to say, it is not possible to cover all potential combinations in a single empirical correlation, and a vast amount of articles regarding how to estimate losses in different junctions exists. In this chapter only some of the most common situations are covered.
When two pipes meet, there are generally going to be losses of four different types: Due to turbulent mixing of two streams moving with different velocities, due to flow turning when it passes from the side branch into the common channel (sometimes enhanced by separation), due to flow expansion in case of diffuser effect or acceleration in case of a nozzle effect, and due to normal pipe friction.
Both Vazsonyi (1944) and Benson et al. (1966) carried out measurements on merging flows of the type seen on figure. 4.7.1. All branches had the same diameter, and the
outlet was perpendicular to the two inlets. The two papers show relatively similar results. When taking the average of the two, we can write the results as:
Figure 4.7.1. Merging flows, all cross-sections equal.
(4.7.1)
As before, the coefficient refers to the velocity in the inlet branch:
(4.7.2)
Due to symmetry, can be computed by simply re-indexing everything so that 1 becomes 2 and vice versa.
For diverging flows, a similar linear estimate of Kf can be obtained by combining the measurements of Vazsoniy (1944) and Benson et al. (1966):
Figure 4.7.2. Diverging flows, all cross-sections equal.
(4.7.3) And:
(4.7.4)
Again, symmetry implies that
can be computed by simple re-indexing to make 1 become 2 and vice versa.
A more general case of merging flows for branches with equal diameters can be extracted from Vazsonyi‟s results (1944). The curves he presented have here been curve-fitted in order to enable easy programming:
(
) [
] (4.7.5)
Figure 4.7.3. General, merging flows, all cross-sections equal
Where again refers to the inlet velocity, and equation 4.7.2 can be used to calculate the pressure loss. The factors:
(4.7.6)
(4.7.7)
(4.7.8)
Vazsonyi‟s results can also be used to estimate pressure losses in diverging flows, but only for 900 T-junctions angled as shown on Figure 4.7.4:
Figure 4.7.4. Diverging flows, all cross-sections equal.
β
c
α
c( ) (
)
(4.7.9)
is as before computed by equation 4.7.7, and by equation 4.7.4.
The last type of branch to be considered here is a pipe of constant diameter receiving fluid from an angled branch of a smaller diameter.
Figure 4.7.5. Merging flows, cross-sections
Idelchik (1992) reports results for many different distinct values of α. Those results may be compressed into the following:
(
)
(
) (4.7.10)
* ( ) ( ) ( ) + (4.7.11)
where k1 is taken from table 4.7.1.
(
) Table 4.7.1. Values of k1.
is a factor which takes the branch angle α (measured in degrees) into account. For varies between 1.7 and 1 in the following way:
(4.7.12)
When calculating , the equation above remains valid all the way up to 900, while something appears to happen with as the angle approaches 900 so that:
(
) (4.7.13)
With these correlations, the pressure loss in each flow-path can then be computed as:
(4.7.14)
(4.7.15)
As we can see, these equations are nicely suited to be included in a computer program.
All data are given in the form of equations, and we may easily write an algorithm which computes both and as a function of for instance ,
and .
To get s feeling for what the correlations express, the loss factors for some angles and cross-sectional areas have been plotted in figures 4.7.6 and 4.7.7:
Figure 4.7.6. as a function of
for α = 300.
Figure 4.7.7. as a function of
for α = 300.
References
Vazsonyi, A. (1944): Pressure loss in elbows and duct branches. Trans ASMA, April, p. 177 Benson, R.S., Wollatt, D. (1966): Compressible flow loss coefficients at bends and T-junctions. Engineer, Vol. 222, January, p. 153.
Benedict, R.P. (1980): Fundamentals of Pipe Flow. John Wiley & Sons.
Crane Technical Paper No. 410 M. (1982): Flow of fluids through valves, fittings and pipe.
Crane Co.
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Idelchik, I.E. (1992): Handbook of Hydraulic Resistance. 3. Edition, Moscow Machine Institute (In Russian). (2. Edition exists in English translation from Hemisphere Publishing Corporation.)
“Blood is a non-Newtonian fluid; its viscosity automatically adjusts to the blood vessel’s diameter.”
Author unknown